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Math Help - Expected value of largest claim on two insured homes (Order Statistics question)

  1. #1
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    Expected value of largest claim on two insured homes (Order Statistics question)

    Suppose two independent calims are made on two insured homes, where each claim has pdf f(x)=4/x^5, 1< x < infinity, in which the unity is $1000. Find the expected value of the largest claim.

    Here was my approach:

    Integrate f(x) to get F(x)=-x^-4
    Apply the order statistics max equation: n{[Fn]^n-1}*f(x) which should have given me: 2(-x^-4)4/x^5. I won't post the computation since the answer is wrong.

    Here is the solution:

    P(X<t) = P(max(x1,x2)<t) ... (i)
    = P(x1<t, x2<t) ... (ii)
    = double integral of fx1,x2(x,y) dx dy (both integrals from -infinity to t) ... (iii)
    = (1-1/t^4)^2, t>1 (iv)

    What I don't get is step (iii) and (iv). How does one go from (ii) to (iii) and then from (iii) to (iv)?

    The rest of the problem is pretty simple. You just plug in the result of P(x<t) into the order statistics min equations and then just take the expected value of that. You end up getting 32/21 for those that care.

    Thanks in advance!
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  2. #2
    MHF Contributor matheagle's Avatar
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    Re: Expected value of largest claim on two insured homes (Order Statistics question)

    How can F(x) be negative? (It can't be)

    F(x)=P(X\le x)=\int_1^x 4t^{-5}dt=1-x^{-4}

    Then explain what you're trying to do, because I'm not sure.
    BTW, I work with order stats all the time...
    Adler : Unusual strong laws for arrays of ratios of order statistics

    http://w3.math.sinica.edu.tw/bulleti...d313/31302.pdf
    And this is a Pareto distribution.
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    Re: Expected value of largest claim on two insured homes (Order Statistics question)

    Thanks for the help Beagle =) Your integral cleared up a lot of issues for me. I was evaluating f(x) from 0 to infinity instead of from 1 to x.

    Basically what I'm trying to do is to get F(x) from f(x), plug that into the order statistics min formula, then integrate that result to get the density function for the min, and then finally take the expectation of that density function.

    Using your integral, I ended up getting the right answer.

    My question now is, why do we evaluate from 1 to x?

    Thanks for the help. I think it's very cool that you work with order statistics often. I used to hate probability and statistics, but over the last year I've grown to like it a bit =)
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    MHF Contributor matheagle's Avatar
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    Re: Expected value of largest claim on two insured homes (Order Statistics question)

    I think you want to know 'why do we evaluate from 1 to x?' in reference to F(x)

    Well F(x)=P(X\le x)=\int_{-\infty}^x f(t)dt by definition.

    In so many setting the min and max are suff statistics, so they need to be studied.
    I have to cover them in my probability class, otherwise in the stat class, I can't assign any problems
    involving densities where the parameters define the support.
    Last edited by matheagle; September 12th 2011 at 03:17 PM.
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